Cou-Cor#

Hide imports 
%load_ext autoreload
%autoreload 2

import itertools
import os
from functools import cache
from typing import List, Literal, Optional, Tuple

import ms3
import pandas as pd
import plotly.express as px
from dimcat import Pipeline, plotting

import utils

pd.set_option("display.max_rows", 1000)
pd.set_option("display.max_columns", 500)
Hide helpers 
RESULTS_PATH = os.path.abspath(os.path.join(utils.OUTPUT_FOLDER, "couperin_study"))
os.makedirs(RESULTS_PATH, exist_ok=True)


def make_output_path(
    filename: str,
    extension=None,
    path=RESULTS_PATH,
) -> str:
    return utils.make_output_path(filename=filename, extension=extension, path=path)


def save_figure_as(
    fig, filename, formats=("png", "pdf"), directory=RESULTS_PATH, **kwargs
):
    if formats is not None:
        for fmt in formats:
            plotting.write_image(fig, filename, directory, format=fmt, **kwargs)
    else:
        plotting.write_image(fig, filename, directory, **kwargs)


def style_plotly(
    fig,
    save_as=None,
    xaxes: Optional[dict] = None,
    yaxes: Optional[dict] = None,
    match_facet_yaxes=False,
    **layout,
):
    layout_args = dict(utils.STD_LAYOUT, **layout)
    fig.update_layout(**layout_args)
    xaxes_settings = dict(gridcolor="lightgrey")
    if xaxes:
        xaxes_settings.update(xaxes)
    fig.update_xaxes(**xaxes_settings)
    yaxes_settings = dict(gridcolor="lightgrey")
    if yaxes:
        yaxes_settings.update(yaxes)
    fig.update_yaxes(**yaxes_settings)
    if match_facet_yaxes:
        for row_idx, row_figs in enumerate(fig._grid_ref):
            for col_idx, col_fig in enumerate(row_figs):
                fig.update_yaxes(
                    row=row_idx + 1,
                    col=col_idx + 1,
                    matches="y" + str(len(row_figs) * row_idx + 1),
                )
    if save_as:
        save_figure_as(fig, save_as)
    fig.show()

Loading data

Hide source 
D = utils.get_dataset("couperin_concerts", corpus_release="v2.2")
D_cor = utils.get_dataset("corelli", corpus_release="v2.7")
D

Dataset
=======
{'inputs': {'basepath': None,
            'packages': {'couperin_concerts': ["'couperin_concerts.measures' (MuseScoreMeasures)",
                                               "'couperin_concerts.notes' (MuseScoreNotes)",
                                               "'couperin_concerts.expanded' (MuseScoreHarmonies)",
                                               "'couperin_concerts.chords' (MuseScoreChords)",
                                               "'couperin_concerts.metadata' (Metadata)"]}},
 'outputs': {'basepath': None, 'packages': {}},
 'pipeline': []}

Grouping data

Hide source 
pipeline = Pipeline(["KeySlicer", "ModeGrouper"])
grouped_D = D.apply_step(pipeline)
grouped_D_cor = D_cor.apply_step(pipeline)
grouped_D

SlicedGroupedDataset
====================
{'inputs': {'basepath': None,
            'packages': {'couperin_concerts': ["'couperin_concerts.measures' (MuseScoreMeasures)",
                                               "'couperin_concerts.notes' (MuseScoreNotes)",
                                               "'couperin_concerts.expanded' (MuseScoreHarmonies)",
                                               "'couperin_concerts.chords' (MuseScoreChords)",
                                               "'couperin_concerts.metadata' (Metadata)"]}},
 'outputs': {'basepath': None,
             'packages': {'features': ["'couperin_concerts.expanded.keyannotations' (KeyAnnotations)"]}},
 'pipeline': ['FeatureExtractor', 'KeySlicer', 'ModeGrouper']}

Starting point: DiMCAT’s BassNotes feature

Hide source 
bass_notes = D.apply_step(pipeline).get_feature("bassnotes")
bass_notes_cor = D_cor.apply_step(pipeline).get_feature("bassnotes")
bass_notes.df
mc mn quarterbeats duration_qb mc_onset mn_onset timesig staff voice volta label pedal chord numeral form figbass changes relativeroot cadence phraseend chord_type chord_tones added_tones root alt_label globalkey_is_minor localkey_is_minor globalkey_mode localkey_mode localkey_resolved localkey_and_mode root_roman relativeroot_resolved effective_localkey effective_localkey_resolved effective_localkey_is_minor pedal_resolved chord_and_mode chord_reduced chord_reduced_and_mode applied_to_numeral numeral_or_applied_to_numeral intervals_over_bass intervals_over_root scale_degrees scale_degrees_and_mode scale_degrees_major scale_degrees_minor bass_degree bass_degree_and_mode bass_degree_major bass_degree_minor bass_note_over_local_tonic globalkey localkey bass_note
mode corpus piece localkey_slice i
major couperin_concerts c01n01_prelude [0.0, 16.0) 0 1 0 0 2.00 0 1/2 4/4 1 1 <NA> G.I{ <NA> I I <NA> <NA> <NA> <NA> <NA> { M (0, 4, 1) () 0 <NA> False False major major I I, major I NaN I I False <NA> I, major I I, major <NA> I (M3, P5) (M3, P5) (1, 3, 5) (1, 3, 5), major (1, 3, 5) (1, #3, 5) 1 1, major 1 1 P1 G I 0
1 2 1 2 2.00 0 0 4/4 1 1 <NA> V <NA> V V <NA> <NA> <NA> <NA> <NA> <NA> M (1, 5, 2) () 1 <NA> False False major major I I, major V NaN I I False <NA> V, major V V, major <NA> V (M3, P5) (M3, P5) (5, 7, 2) (5, 7, 2), major (5, 7, 2) (5, #7, 2) 5 5, major 5 5 P5 G I 1
2 2 1 4 0.50 1/2 1/2 4/4 1 1 <NA> I6 <NA> I6 I <NA> 6 <NA> <NA> <NA> <NA> M (4, 1, 0) () 0 <NA> False False major major I I, major I NaN I I False <NA> I6, major I6 I6, major <NA> I (m3, m6) (M3, P5) (3, 5, 1) (3, 5, 1), major (3, 5, 1) (#3, 5, 1) 3 3, major 3 #3 M3 G I 4
3 2 1 9/2 0.50 5/8 5/8 4/4 1 1 <NA> I <NA> I I <NA> <NA> <NA> <NA> <NA> <NA> M (0, 4, 1) () 0 <NA> False False major major I I, major I NaN I I False <NA> I, major I I, major <NA> I (M3, P5) (M3, P5) (1, 3, 5) (1, 3, 5), major (1, 3, 5) (1, #3, 5) 1 1, major 1 1 P1 G I 0
4 2 1 5 0.75 3/4 3/4 4/4 1 1 <NA> V(4) <NA> V(4) V <NA> <NA> 4 <NA> <NA> <NA> M (1, 0, 2) () 1 <NA> False False major major I I, major V NaN I I False <NA> V(4), major V V, major <NA> V (P4, P5) (P4, P5) (5, 1, 2) (5, 1, 2), major (5, 1, 2) (5, 1, 2) 5 5, major 5 5 P5 G I 1
... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
minor couperin_concerts parnasse_07 [173.0, 212.0) 230 52 52 411/2 0.25 3/8 3/8 4/4 1 1 <NA> i64 <NA> i64 i <NA> 64 <NA> <NA> <NA> <NA> m (1, 0, -3) () 0 <NA> True True minor minor i i, minor i NaN i i True <NA> i64, minor i64 i64, minor <NA> i (P4, m6) (m3, P5) (5, 1, 3) (5, 1, 3), minor (5, 1, b3) (5, 1, 3) 5 5, minor 5 5 P5 b i 1
231 52 52 823/4 0.25 7/16 7/16 4/4 1 1 <NA> iio64 <NA> iio64 ii o 64 <NA> <NA> <NA> <NA> o (-4, 2, -1) () 2 <NA> True True minor minor i i, minor ii NaN i i True <NA> iio64, minor iio64 iio64, minor <NA> ii (a4, M6) (m3, d5) (6, 2, 4) (6, 2, 4), minor (b6, 2, 4) (6, 2, 4) 6 6, minor b6 6 m6 b i -4
232 52 52 206 1.00 1/2 1/2 4/4 1 1 <NA> i6 <NA> i6 i <NA> 6 <NA> <NA> <NA> <NA> m (-3, 1, 0) () 0 <NA> True True minor minor i i, minor i NaN i i True <NA> i6, minor i6 i6, minor <NA> i (M3, M6) (m3, P5) (3, 5, 1) (3, 5, 1), minor (b3, 5, 1) (3, 5, 1) 3 3, minor b3 3 m3 b i -3
233 52 52 207 1.00 3/4 3/4 4/4 1 1 <NA> V <NA> V V <NA> <NA> <NA> <NA> <NA> <NA> M (1, 5, 2) () 1 <NA> True True minor minor i i, minor V NaN i i True <NA> V, minor V V, minor <NA> V (M3, P5) (M3, P5) (5, #7, 2) (5, #7, 2), minor (5, 7, 2) (5, #7, 2) 5 5, minor 5 5 P5 b i 1
234 53 53 208 4.00 0 0 4/4 1 1 <NA> i|PAC} <NA> i i <NA> <NA> <NA> <NA> PAC } m (0, -3, 1) () 0 <NA> True True minor minor i i, minor i NaN i i True <NA> i, minor i i, minor <NA> i (m3, P5) (m3, P5) (1, 3, 5) (1, 3, 5), minor (1, b3, 5) (1, 3, 5) 1 1, minor 1 1 P1 b i 0

8376 rows × 56 columns

If needed, the localkey_slice intervals can be resolved using this table:

Hide source 
local_keys = grouped_D.get_feature("KeyAnnotations")
utils.print_heading("Key Segments Couperin")
print(local_keys.groupby("mode").size().to_string())
local_keys_cor = grouped_D_cor.get_feature("KeyAnnotations")
utils.print_heading("\nKey Segments Corelli")
print(local_keys_cor.groupby("mode").size().to_string())
local_keys.head()

Key Segments Couperin
---------------------

mode
major    279
minor    287

Key Segments Corelli
---------------------

mode
major    345
minor    367
mc mn quarterbeats duration_qb mc_onset mn_onset timesig staff voice volta label globalkey_is_minor localkey_is_minor globalkey_mode localkey_mode localkey_resolved localkey_and_mode globalkey localkey
mode corpus piece localkey_slice i
major couperin_concerts c01n01_prelude [0.0, 16.0) 0 1 0 0 16.0 0 1/2 4/4 1 1 <NA> G.I{ False False major major I I, major G I
[22.5, 32.0) 22 7 6 45/2 9.5 1/8 1/8 4/4 1 1 <NA> V.V{ False False major major V V, major G V
[32.0, 40.0) 35 9 8 32 8.0 1/2 1/2 4/4 1 1 <NA> IV.ii6{ False False major major IV IV, major G IV
[40.0, 47.0) 41 11 9 40 7.0 1/2 1/2 4/4 1 1 <NA> V.V{ False False major major V V, major G V
[47.0, 98.0) 48 13 11 47 51.0 1/4 1/4 4/4 1 1 <NA> I.V65 False False major major I I, major G I
Hide helpers 
succession_map = dict(
    ascending_major={
        "1": "2",
        "2": "3",
        "3": "4",
        "4": "5",
        "5": "6",
        "6": "7",
        "7": "1",
    },
    ascending_minor={
        "1": "2",
        "2": "3",
        "3": "4",
        "4": "5",
        "5": "#6",
        "#6": "#7",
        "#7": "1",
    },
    descending={"1": "7", "2": "1", "3": "2", "4": "3", "5": "4", "6": "5", "7": "6"},
)


def inverse_dict(d):
    return {v: k for k, v in d.items()}


predecessor_map = dict(
    ascending_major=inverse_dict(succession_map["ascending_major"]),
    ascending_minor=inverse_dict(succession_map["ascending_minor"]),
    descending=inverse_dict(succession_map["descending"]),
)


def make_precise_preceding_movement_column(df):
    """Expects a dataframe containing the columns bass_degree, preceding_bass_degree, and preceding_movement,"""
    preceding_movement_precise = df.preceding_movement.where(
        df.preceding_movement != "step", df.preceding_interval
    )
    expected_ascending_degree = pd.concat(
        [
            df.loc[["major"], "bass_degree"].map(predecessor_map["ascending_major"]),
            df.loc[["minor"], "bass_degree"].map(predecessor_map["ascending_minor"]),
        ]
    )
    expected_descending_degree = df.bass_degree.map(predecessor_map["descending"])
    preceding_movement_precise = preceding_movement_precise.where(
        df.preceding_bass_degree != expected_ascending_degree, "ascending"
    )
    preceding_movement_precise = preceding_movement_precise.where(
        df.preceding_bass_degree != expected_descending_degree, "descending"
    )
    return preceding_movement_precise


def make_precise_subsequent_movement_column(df):
    """Expects a dataframe containing the columns bass_degree, subsequent_bass_degree, and subsequent_movement,"""
    subsequent_movement_precise = df.subsequent_movement.where(
        df.subsequent_movement != "step", df.subsequent_interval
    )
    expected_ascending_degree = pd.concat(
        [
            df.loc[["major"], "bass_degree"].map(succession_map["ascending_major"]),
            df.loc[["minor"], "bass_degree"].map(succession_map["ascending_minor"]),
        ]
    )
    expected_descending_degree = df.bass_degree.map(succession_map["descending"])
    subsequent_movement_precise = subsequent_movement_precise.where(
        df.subsequent_bass_degree != expected_ascending_degree, "ascending"
    )
    subsequent_movement_precise = subsequent_movement_precise.where(
        df.subsequent_bass_degree != expected_descending_degree, "descending"
    )
    return subsequent_movement_precise

This is the main table of this notebook. It corresponds to the BassNotes features, with a preceding_ and a subsequent_ copy of each column concatenated to the right. The respective upward and downward shifts are performed within each localkey group, leaving first bass degrees with undefined preceding values and last bass degrees without undefined subsequent values.

Hide source 
def make_adjacency_table(bass_notes):
    preceding = bass_notes.groupby(["piece", "localkey_slice"]).shift()
    preceding.columns = "preceding_" + preceding.columns
    subsequent = bass_notes.groupby(["piece", "localkey_slice"]).shift(-1)
    subsequent.columns = "subsequent_" + subsequent.columns
    BN = pd.concat([bass_notes, preceding, subsequent], axis=1)
    BN["preceding_iv"] = BN.bass_note - BN.preceding_bass_note
    BN["subsequent_iv"] = BN.subsequent_bass_note - BN.bass_note
    BN["preceding_interval"] = ms3.transform(
        BN.preceding_iv, ms3.fifths2iv, smallest=True
    )
    BN["subsequent_interval"] = ms3.transform(
        BN.subsequent_iv, ms3.fifths2iv, smallest=True
    )
    BN["preceding_iv_is_step"] = BN.preceding_iv.isin(
        (-5, -2, 2, 5)
    ).where(  # +m2, -M2, +M2, -m2
        BN.preceding_iv.notna()
    )
    BN["subsequent_iv_is_step"] = BN.subsequent_iv.isin((-5, -2, 2, 5)).where(
        BN.subsequent_iv.notna()
    )
    BN["preceding_iv_is_0"] = BN.preceding_iv == 0
    BN["subsequent_iv_is_0"] = BN.subsequent_iv == 0
    BN["preceding_movement"] = (
        BN.preceding_iv_is_step.map({True: "step", False: "leap"})
        .where(~BN.preceding_iv_is_0, "same")
        .where(BN.preceding_iv.notna(), "none")
    )
    BN["subsequent_movement"] = (
        BN.subsequent_iv_is_step.map({True: "step", False: "leap"})
        .where(~BN.subsequent_iv_is_0, "same")
        .where(BN.subsequent_iv.notna(), "none")
    )
    BN["preceding_movement_precise"] = make_precise_preceding_movement_column(BN)
    BN["subsequent_movement_precise"] = make_precise_subsequent_movement_column(BN)
    return BN


BN = make_adjacency_table(bass_notes)
BN_cor = make_adjacency_table(bass_notes_cor)
Hide source 
ignore_mask = BN.subsequent_interval.isna() | BN.subsequent_interval.duplicated()
interval2fifths = (  # mapping that allows to order the x-axis with intervals according to LoF
    BN.loc[~ignore_mask, ["subsequent_interval", "subsequent_iv"]]
    .set_index("subsequent_interval")
    .iloc[:, 0]
    .sort_values()
)

Overview of how the bass moves#

Intervals#

Hide source 
def plot_bass_movement(BN, corpus_name):
    interval_data = pd.concat(
        [
            BN.groupby("mode").subsequent_interval.value_counts(normalize=True),
            BN.groupby(["piece", "mode"])
            .subsequent_interval.value_counts(normalize=True)
            .groupby(["mode", "subsequent_interval"])
            .sem()
            .rename("std_err"),
        ],
        axis=1,
    ).reset_index()
    fig = px.bar(
        interval_data,
        x="subsequent_interval",
        y="proportion",
        color="mode",
        barmode="group",
        error_y="std_err",
        color_discrete_map=utils.MAJOR_MINOR_COLORS,
        labels=dict(subsequent_interval="Interval"),
        title=f"Mode-wise proportion of how often a bass note moves by an interval in {corpus_name}",
        category_orders=dict(subsequent_interval=interval2fifths.index),
    )
    style_plotly(fig, f"how_often_a_bass_note_moves_by_an_interval_{corpus_name}")


plot_bass_movement(BN, "Couperin")
plot_bass_movement(BN_cor, "Corelli")

Types of movement#

The values ascending and descending designate stepwise movement within the regola. Only non-chromatic scale degrees can have these values with the exception of #6 and #7 which are considered diatonic in the context of this study.

Hide source 
def plot_movement_types(BN, corpus_name, precise_categories=True):
    subsequent_movement = (
        "subsequent_movement_precise" if precise_categories else "subsequent_movement"
    )
    movement_data = pd.concat(
        [
            BN.groupby("mode")[subsequent_movement].value_counts(
                normalize=True, dropna=False
            ),
            BN.groupby(["piece", "mode"])[subsequent_movement]
            .value_counts(normalize=True, dropna=False)
            .groupby(["mode", subsequent_movement])
            .sem()
            .rename("std_err"),
        ],
        axis=1,
    ).reset_index()
    movement_data[subsequent_movement] = movement_data[subsequent_movement].fillna(
        "none"
    )
    fig = px.bar(
        movement_data,
        x=subsequent_movement,
        y="proportion",
        color="mode",
        barmode="group",
        error_y="std_err",
        color_discrete_map=utils.MAJOR_MINOR_COLORS,
        labels={subsequent_movement: "Movement"},
        title=f"Mode-wise proportion of a bass note moving in a certain manner in {corpus_name}",
        category_orders=dict(subsequent_interval=interval2fifths.index),
    )
    style_plotly(fig, save_as=f"mode-wise_bass_motion_{corpus_name}")


plot_movement_types(BN, "Couperin")
plot_movement_types(BN_cor, "Corelli")

Sankey diagrams showing movement types before and after each scale degree#

Hide helpers 
def make_sankey_data(
    five_major, color_edges=True, precise=True
) -> Tuple[pd.DataFrame, List[str], List[str]] | Tuple[pd.DataFrame, List[str]]:
    preceding_movement = (
        "preceding_movement_precise" if precise else "preceding_movement"
    )
    subsequent_movement = (
        "subsequent_movement_precise" if precise else "subsequent_movement"
    )
    type_counts = five_major["intervals_over_bass"].value_counts()
    preceding_movement_counts = five_major[preceding_movement].value_counts()
    subsequent_movement_counts = five_major[subsequent_movement].value_counts()
    preceding_links = five_major.groupby(
        [preceding_movement]
    ).intervals_over_bass.value_counts()
    subsequent_links = five_major.groupby(
        [subsequent_movement]
    ).intervals_over_bass.value_counts()

    node_labels = []
    label_ids = dict()
    for key, node_sizes in (
        ("preceding", preceding_movement_counts),
        ("intervals", type_counts),
        ("subsequent", subsequent_movement_counts),
    ):
        for label in node_sizes.index:
            label_id = len(node_labels)
            node_labels.append(str(label))
            label_ids[(key, label)] = label_id

    edge_columns = ["source", "target", "value"]
    if color_edges:
        node_colors = utils.make_evenly_distributed_color_map(node_labels)
        edge_columns.append("color")

    links = []
    for (prec_mov, iv), cnt in preceding_links.items():
        source_id = label_ids.get(("preceding", prec_mov))
        target_id = label_ids.get(("intervals", iv))
        if color_edges:
            edge_color = node_colors[source_id]
            links.append((source_id, target_id, cnt, edge_color))
        else:
            links.append((source_id, target_id, cnt))

    for (subs_mov, iv), cnt in subsequent_links.items():
        source_id = label_ids.get(("intervals", iv))
        target_id = label_ids.get(("subsequent", subs_mov))
        if color_edges:
            edge_color = node_colors[target_id]
            links.append((source_id, target_id, cnt, edge_color))
        else:
            links.append((source_id, target_id, cnt))

    edge_data = pd.DataFrame(links, columns=edge_columns)
    if color_edges:
        return edge_data, node_labels, node_colors
    return edge_data, node_labels


def make_bass_degree_sankey(
    BN, corpus, bass_degree: str, mode: Literal["major", "minor"], **layout
):
    edge_data, node_labels, node_colors = make_sankey_data(
        BN.loc[mode].query(f"bass_degree == '{bass_degree}'")
    )
    title = f"Motions to and from bass degree {bass_degree} in {corpus}"
    fig = utils.make_sankey(
        edge_data, node_labels, node_color=node_colors, title=title, **layout
    )
    return fig

Intervals over bass degree 1#

Major#

Hide source 
make_bass_degree_sankey(BN, "Couperin", 1, "major")
make_bass_degree_sankey(BN_cor, "Corelli", 1, "major")

Minor#

Hide source 
make_bass_degree_sankey(BN, "Couperin", 1, "minor")
make_bass_degree_sankey(BN_cor, "Corelli", 1, "minor")

Intervals over bass degree 2#

Major#

make_bass_degree_sankey(BN, "Couperin", 2, "major")
make_bass_degree_sankey(BN_cor, "Corelli", 2, "major")

Minor#

make_bass_degree_sankey(BN, "Couperin", 2, "minor")
make_bass_degree_sankey(BN_cor, "Corelli", 2, "minor")

Intervals over bass degree 3#

Major#

make_bass_degree_sankey(BN, "Couperin", 3, "major")
make_bass_degree_sankey(BN_cor, "Corelli", 3, "major")

Minor#

make_bass_degree_sankey(BN, "Couperin", 3, "minor")
make_bass_degree_sankey(BN_cor, "Corelli", 3, "minor")

Intervals over bass degree 4#

Major#

make_bass_degree_sankey(BN, "Couperin", 4, "major")
make_bass_degree_sankey(BN_cor, "Corelli", 4, "major")

Minor#

make_bass_degree_sankey(BN, "Couperin", 4, "minor")
make_bass_degree_sankey(BN_cor, "Corelli", 4, "minor")

Intervals over bass degree 5#

Major#

make_bass_degree_sankey(BN, "Couperin", 5, "major")
make_bass_degree_sankey(BN_cor, "Corelli", 5, "major")

Minor#

make_bass_degree_sankey(BN, "Couperin", 5, "minor")
make_bass_degree_sankey(BN_cor, "Corelli", 5, "minor")

Intervals over bass degree 6#

Major#

make_bass_degree_sankey(BN, "Couperin", 6, "major")
make_bass_degree_sankey(BN_cor, "Corelli", 6, "major")

Minor (ascending)#

make_bass_degree_sankey(BN, "Couperin", "#6", "minor")
make_bass_degree_sankey(BN, "Corelli", "#6", "minor")

Minor (descending)#

make_bass_degree_sankey(BN, "Couperin", 6, "minor")
make_bass_degree_sankey(BN_cor, "Corelli", 6, "minor")

Intervals over bass degree 7#

Major#

make_bass_degree_sankey(BN, "Couperin", 7, "major")
make_bass_degree_sankey(BN_cor, "Corelli", 7, "major")

Minor (ascending)#

make_bass_degree_sankey(BN, "Couperin", "#7", "minor")
make_bass_degree_sankey(BN, "Corelli", "#7", "minor")

Minor (descending)#

make_bass_degree_sankey(BN, "Couperin", 7, "minor")
make_bass_degree_sankey(BN_cor, "Corelli", 7, "minor")

Explanatory power of the RoO#

Most frequent chord for each bass degree#

Couperin#

Hide source 
BN.groupby(["mode", "bass_degree"]).intervals_over_bass.apply(
    lambda S: S.value_counts().idxmax()
)

mode   bass_degree
major  #1             (m3, d5, m6)
       #4             (m3, d5, m6)
       #5             (m3, d5, d7)
       1                  (M3, P5)
       2              (m3, P4, M6)
       3                  (m3, m6)
       4                  (M3, P5)
       5                  (M3, P5)
       6                  (m3, P5)
       7              (m3, d5, m6)
       b3             (M3, a5, M7)
       b7             (M2, a4, M6)
minor  #3             (m3, d5, m6)
       #4             (m3, d5, m6)
       #6                 (m3, m6)
       #7             (m3, d5, m6)
       1                  (m3, P5)
       2              (m3, P4, M6)
       3                  (M3, M6)
       4              (m3, P5, M6)
       5                  (M3, P5)
       6              (M3, a4, M6)
       7                  (M3, M6)
Name: intervals_over_bass, dtype: object

Couperin#

BN_cor.groupby(["mode", "bass_degree"]).intervals_over_bass.apply(
    lambda S: S.value_counts().idxmax()
)

mode   bass_degree
major  #1             (m3, d5, m6)
       #2                 (m3, m6)
       #4                 (m3, m6)
       #5                 (m3, m6)
       1                  (M3, P5)
       2                  (m3, P5)
       3                  (m3, m6)
       4                  (M3, P5)
       5                  (M3, P5)
       6                  (m3, P5)
       7                  (m3, m6)
       b2                 (M3, M6)
       b3                 (M3, M6)
       b6                 (M3, M6)
       b7                 (M3, P5)
minor  #3                 (m3, m6)
       #4             (m3, d5, d7)
       #6                 (m3, m6)
       #7                 (m3, m6)
       1                  (m3, P5)
       2                  (m3, M6)
       3                  (M3, M6)
       4                  (m3, P5)
       5                  (M3, P5)
       6                  (M3, M6)
       7                  (M3, M6)
       b1             (M3, P5, M6)
       b2                 (m3, M6)
       b5                 (M3, M6)
       b7                 (a4, M6)
Name: intervals_over_bass, dtype: object
Hide source 
maj = ("M3", "P5")
maj6 = ("m3", "m6")
min = ("m3", "P5")
min6 = ("M3", "M6")
Mm56 = ("m3", "d5", "m6")
Mm34 = ("m3", "P4", "M6")
Mm24 = ("M2", "a4", "M6")
mm56 = ("M3", "P5", "M6")
hdim56 = ("m3", "P5", "M6")
hdim34 = ("M3", "a4", "M6")

regole = dict(
    ascending_major=[
        ("1", maj),  # most frequent
        ("2", Mm34),  # most frequent
        ("3", maj6),  # most frequent
        ("4", mm56),  # not most frequent
        ("5", maj),  # most frequent
        ("6", min6),  # not most frequent
        ("7", Mm56),  # most frequent
    ],
    descending_major=[
        ("1", maj),  # same
        ("7", maj6),  # different, not most frequent
        ("6", Mm34),  # different, not most frequent either
        ("5", maj),  # same
        ("4", Mm24),  # different, not most frequent either
        ("3", maj6),  # same
        ("2", Mm34),  # same
    ],
    ascending_minor=[
        ("1", min),  # most frequent
        ("2", Mm34),  # most frequent
        ("3", min6),  # most frequent
        ("4", hdim56),  # most frequent
        ("5", maj),  # most frequent
        ("#6", maj6),  # most frequent
        ("#7", Mm56),  # most frequent
    ],
    descending_minor=[
        ("1", min),  # same
        ("7", min6),  # different, most frequent
        ("6", hdim34),  # different, most frequent
        ("5", maj),  # same
        ("4", Mm24),  # different, not most frequent
        ("3", min6),  # same
        ("2", Mm34),  # same
    ],
)
Hide helpers 
@cache
def get_base_df(
    basis: Literal[
        "major_all", "minor_all", "major_diatonic", "minor_diatonic"
    ],  # minor_diatonic includes 6, #6, 7, #7
    query: Optional[str] = None,
):
    global BN
    try:
        mode, selection = basis.split("_")
    except Exception:
        raise ValueError(f"Invalid keyword for basis: {basis!r}")
    base = BN.loc[[mode]]
    if selection == "all":
        result = base
    elif selection == "diatonic":
        if mode == "major":
            result = base.query("bass_degree in ('1', '2', '3', '4', '5', '6', '7')")
        elif mode == "minor":
            result = base.query(
                "bass_degree in ('1', '2', '3', '4', '5', '6', '#6', '7', '#7')"
            )
    else:
        raise ValueError(f"Unknown keyword for selection: {selection!r}")
    if query:
        result = result.query(query)
    return result


@cache
def get_bass_degree_mask(
    basis: Literal[
        "major_all", "minor_all", "major_diatonic", "minor_diatonic"
    ],  # minor_diatonic includes 6, #6, 7, #7
    bass_degree: str,
    query: Optional[str] = None,
):
    base = get_base_df(basis, query=query)
    return base.bass_degree == bass_degree


@cache
def get_intervals_mask(
    basis: Literal[
        "major_all", "minor_all", "major_diatonic", "minor_diatonic"
    ],  # minor_diatonic includes 6, #6, 7, #7
    intervals: tuple,
    query: Optional[str] = None,
):
    base = get_base_df(basis, query=query)
    return base.intervals_over_bass == intervals


@cache
def get_chord_mask(
    basis: Literal[
        "major_all", "minor_all", "major_diatonic", "minor_diatonic"
    ],  # minor_diatonic includes 6, #6, 7, #7
    bass_degree: str,
    intervals: tuple,
    query: Optional[str] = None,
):
    bass_degree_mask = get_bass_degree_mask(
        basis=basis, bass_degree=bass_degree, query=query
    )
    intervals_mask = get_intervals_mask(basis=basis, intervals=intervals, query=query)
    return bass_degree_mask & intervals_mask


@cache
def get_chord_vocabulary_mask(
    basis: Literal[
        "major_all", "minor_all", "major_diatonic", "minor_diatonic"
    ],  # minor_diatonic includes 6, #6, 7, #7
    vocabulary: Tuple[Tuple[str, tuple], ...],
    query: Optional[str] = None,
) -> pd.Series:
    base = get_base_df(basis, query=query)
    mask = pd.Series(False, index=base.index, dtype="boolean")
    for bass_degree, intervals in vocabulary:
        mask |= get_chord_mask(
            basis=basis, bass_degree=bass_degree, intervals=intervals, query=query
        )
    return mask


def inspect(
    basis: Literal[
        "major_all", "minor_all", "major_diatonic", "minor_diatonic"
    ],  # minor_diatonic includes 6, #6, 7, #7
    vocabulary: Tuple[Tuple[str, tuple], ...],
    query: Optional[str] = None,
) -> pd.DataFrame:
    base = get_base_df(basis, query=query)
    mask = get_chord_vocabulary_mask(basis=basis, vocabulary=vocabulary, query=query)
    return base[mask]


def get_vocabulary_coverage(
    basis: Literal[
        "major_all", "minor_all", "major_diatonic", "minor_diatonic"
    ],  # minor_diatonic includes 6, #6, 7, #7
    vocabulary: Tuple[Tuple[str, tuple], ...],
    query: Optional[str] = None,
) -> float:
    mask = get_chord_vocabulary_mask(basis=basis, vocabulary=vocabulary, query=query)
    return mask.sum() / len(mask)


def get_coverage_values(
    major_vocabulary: Optional[Tuple[Tuple[str, tuple], ...]] = None,
    minor_vocabulary: Optional[Tuple[Tuple[str, tuple], ...]] = None,
    **name2query,
) -> pd.Series:
    if not (major_vocabulary or minor_vocabulary):
        return pd.Series()
    results = {}
    if major_vocabulary:
        results.update(
            {
                ("major", "all"): get_vocabulary_coverage(
                    "major_all", major_vocabulary
                ),
                ("major", "diatonic"): get_vocabulary_coverage(
                    "major_diatonic", major_vocabulary
                ),
            }
        )
        for name, query in name2query.items():
            results[("major", name)] = get_vocabulary_coverage(
                "major_diatonic", major_vocabulary, query=query
            )
    if minor_vocabulary:
        results.update(
            {
                ("minor", "all"): get_vocabulary_coverage(
                    "minor_all", minor_vocabulary
                ),
                ("minor", "diatonic"): get_vocabulary_coverage(
                    "minor_diatonic", minor_vocabulary
                ),
            }
        )
        for name, query in name2query.items():
            results[("minor", name)] = get_vocabulary_coverage(
                "minor_diatonic", minor_vocabulary, query=query
            )
    result = pd.Series(results, name="proportion")
    result.index.names = ["mode", "coverage_of"]
    return result

Which proportion of unigrams are “explained” by Campion’s regola#

The percentages are based on different sets of unigrams. from means before/leading to a bass degree, to means after/following a bass degree.

  • all: all bass degrees

  • diatonic: all non-chromatic bass degrees (in minor, the chromatic scale degrees #6 and #7 are considered diatonic)

  • to_ascending: all diatonic bass degrees that ascend within the regola

  • from_ascending: all diatonic bass degrees that are reached by ascending within the regola

  • to_and_from_ascending: all diatonic bass degrees that are reached by ascending within the regola and proceed ascending within the regola

  • to_and_from_either: all diatonic bass degrees whose predecessor and successor are both upper or lower neighbors within the regola

  • to_leap: all diatonic bass degrees followed by a leap

  • to_same: all diatonic bass degrees followed by the same bass degree

  • etc.

Hide source 
regola_vocabulary_major = tuple(
    set(regole["ascending_major"] + regole["descending_major"])
)
regola_vocabulary_minor = tuple(
    set(regole["ascending_minor"] + regole["descending_minor"])
)

features = dict(
    to_ascending="subsequent_movement_precise == 'ascending'",
    to_descending="subsequent_movement_precise == 'descending'",
    to_either="subsequent_movement_precise == ['ascending', 'descending']",
    to_leap="subsequent_movement == 'leap'",
    to_same="subsequent_movement == 'same'",
    last_notes="subsequent_movement == 'none'",
    from_ascending="preceding_movement_precise == 'ascending'",
    from_descending="preceding_movement_precise == 'descending'",
    from_either="preceding_movement_precise == ['ascending', 'descending']",
    from_leap="preceding_movement == 'leap'",
    from_same="preceding_movement == 'same'",
    first_notes="preceding_movement == 'none'",
    to_and_from_ascending="subsequent_movement_precise == 'ascending' & preceding_movement_precise == 'ascending'",
    to_and_from_descending="subsequent_movement_precise == 'descending' & preceding_movement_precise == 'descending'",
    to_and_from_either="subsequent_movement_precise == ['ascending', 'descending'] & "
    "preceding_movement_precise == ['ascending', 'descending']",
    to_and_from_leap="subsequent_movement == 'leap' & preceding_movement == 'leap'",
    to_and_from_same="subsequent_movement == 'same' & preceding_movement == 'same'",
)

regola_coverage = get_coverage_values(
    regola_vocabulary_major, regola_vocabulary_minor, **features
)
regola_coverage

mode   coverage_of           
major  all                       0.651522
       diatonic                  0.662083
       to_ascending              0.738881
       to_descending             0.723051
       to_either                 0.731538
       to_leap                   0.652388
       to_same                   0.403084
       last_notes                0.827957
       from_ascending            0.680057
       from_descending           0.787728
       from_either               0.730000
       from_leap                 0.665669
       from_same                 0.436123
       first_notes               0.726277
       to_and_from_ascending     0.768293
       to_and_from_descending    0.852814
       to_and_from_either        0.824176
       to_and_from_leap          0.624314
       to_and_from_same          0.444444
minor  all                       0.626947
       diatonic                  0.638242
       to_ascending              0.765468
       to_descending             0.714504
       to_either                 0.740741
       to_leap                   0.643806
       to_same                   0.363229
       last_notes                0.629371
       from_ascending            0.705036
       from_descending           0.714504
       from_either               0.709630
       from_leap                 0.643243
       from_same                 0.461883
       first_notes               0.684783
       to_and_from_ascending     0.811321
       to_and_from_descending    0.745174
       to_and_from_either        0.783835
       to_and_from_leap          0.620690
       to_and_from_same          0.409091
Name: proportion, dtype: float64

Comparing the regola against all “top k” vocabularies#

Campion’s regola comprises 10 different chords for both major and minor. For comparison, its values are shown at point 10.5 on the x-axis. The lower two plots show how many unigrams are covered by individual chords. Hover over the points to see the corresponding chords.

Hide helpers 
def make_coverage_plot_data(
    include_singular_vocabularies=True, **features
) -> pd.DataFrame:
    all_chords = BN[["bass_degree", "intervals_over_bass"]].apply(tuple, axis=1)
    chord_ranking = all_chords.groupby("mode").value_counts(normalize=True)
    major_ranking, minor_ranking = (
        chord_ranking.loc["major"],
        chord_ranking.loc["minor"],
    )
    major_vocab, minor_vocab = [], []
    results = {}
    for i, (maj_chord, min_chord) in enumerate(
        itertools.zip_longest(major_ranking.index, minor_ranking.index), 1
    ):
        if maj_chord:
            major_vocab.append(maj_chord)
        if min_chord:
            minor_vocab.append(min_chord)
        key = ("cumulative", i) if include_singular_vocabularies else i
        values = get_coverage_values(tuple(major_vocab), tuple(minor_vocab), **features)
        chord = pd.Series(str(maj_chord), index=values.index, name="chord")
        chord.loc["minor"] = str(min_chord)
        results[key] = pd.concat([values, chord], axis=1)
        if not include_singular_vocabularies:
            continue
        single_maj_vocab = (maj_chord,) if maj_chord else None
        single_min_vocab = (min_chord,) if min_chord else None
        values = get_coverage_values(single_maj_vocab, single_min_vocab, **features)
        results[("single", i)] = pd.concat([values, chord], axis=1)
    index_levels = ["vocabulary", "rank"] if include_singular_vocabularies else ["rank"]
    return pd.concat(results, names=index_levels)
Hide source 
result = make_coverage_plot_data(**features)
regola_results = pd.concat(
    {("cumulative", 10.5): regola_coverage}, names=["vocabulary", "rank"]
).to_frame()
regola_results.loc[:, "chord"] = "regola"
result = pd.concat(
    [
        regola_results,
        result,
    ]
).sort_index()
Hide source 
fig = px.line(
    result.reset_index(),
    x="rank",
    y="proportion",
    color="coverage_of",
    facet_col="mode",
    facet_row="vocabulary",
    hover_name="chord",
    log_x=True,
    title="How many unigrams are covered by each top-k vocabulary",
)
style_plotly(
    fig,
    match_facet_yaxes=True,
    height=1500,
    legend=dict(
        orientation="h",
    ),
)

In order to inspect these plots you will want to hide traces. Click on a legend item to toggle it, double-click on an item to toggle all others.

In order to inspect these plots you will want to hide traces. Click on a legend item to toggle it, double-click on an item to toggle all others.